5. Weak and weak* topologies
Definition 5.1. Let Zbe a set and let (Zi,Ti)i∈Ibe a family of topological spaces. For each ilet pi:Z→Zibe a mapping. We
define the initial topology on Z(associated with the mappings pi) to be the coarsest topology on Zfor which all the mappings pi
are continuous. Its basic open sets are of the form
\
i∈F
p−1
i[Ui],
with Fa finite subset of Iand Ui∈Tifor each i.
There are many important examples. For instance, given the family (Zi)i∈Ias above, we may consider the product space
Z=Y
i∈I
Zi,
whose elements are all functions fthat assign to each i∈Ian element f(i) of Zi. (We sometimes prefer to use “family” notation
(zi)i∈Ifor such things, in order to emphasize the similarity with sequences and the idea that f(i) is the “ith co-ordinate” of f.)
If all the spaces Ziare copies of a single space Z, we write ZIfor the product space. On the product space Zwe introduce the
co-ordinate mappings
πi:Z→Zi;f7→ f(i),
and define the (Tychonoff) product topology to be the initial topology with respect to the mappings πi. It is an important theorem
(Tychonoff’s Theorem) that if all the spaces Ziare compact and Hausdorff, then so is Z. This theorem is presented in the C1
Analytic Topology course.
If Lis a topological space then the space of all continuous real-valued functions C(L) is a subspace of RL. The topology induced
by the product topology is the topology of pointwise convergence, basis open neighbourhoods of a function fbeing of the form
{y∈C(L) : |y(ti)−x(ti)|< for i= 1,2, . . . , n},
where nis a natural number, > 0 and t1, . . . , tn∈L.
Let Xbe a real vector space and let Ybe a subspace of the algebraic dual X#. We define the topology σ(X, Y ) to be the
coarsest topology on Xfor which all the maps g:X→R(g∈Y) are continuous. Basis open sets have the form
{x0∈X:|gi(x0)−gi(x)|< for i= 1,2, . . . , n}.
In the important case where Xis a normed space and Y=X∗we call this topology the weak topology on X.
When Xis a normed space, we can also introduce a topology on X∗, defined to be the coarsest topology for which all the
mappings f7→ f(x) (x∈X) are continuous. This is called the weak* topology on X∗. In the σ-notation, this topology is just
σ(X∗, JX[X]) (or, less pedantically, σ(X∗, X). Notice that the weak topology on Xis coarser than the norm topology (because
all f∈X∗are norm-continuous). In Proposition 5.4 we show that the weak topology is strictly coarser than the norm topology
whenever Xis infinite-dimensional. Note also that the weak* topology on X∗is coarser (and in the non-reflexive case strictly
coarser) than the weak topology σ(X∗, X∗∗ ).
Lemma 5.2. If Xis a real vector space and Yis a subspace of X#then every σ(X,Y )-open neighbourhood of 0contains a
finite-codimensional subspace X1of X.
Proof. A basic σ(X, Y )-open neighbourhood of 0 is of the form
{x∈X:|gj(x)|< ifor i= 1,2, . . . , n}
where gi∈Yand i>0. This set contains the finite-codimensional subspace
X1=\
i≤n
ker gi.
Proposition 5.3. If Xis a real vector space and Yis a subspace of X#then a linear mapping g:X→Ris σ(X, Y )-continuous
if and only if g∈Y.
Proof. If g∈Ythen gis σ(X, Y )-continuous by definition of the topology. On the other hand, if gis linear and σ(X , Y )-continuous
then there exists a σ(X, Y )-open neighbourhood Uof 0 such that |g(x)| ≤ 1 for all x∈U. It follows that gis zero on the subspace
X1=Ti≤nker giof Lemma 5.2. It follows “by algebra” that gis a linear combination Pi≤nλigiand hence in the subspace Y.
Proposition 5.4. If Xis an infinite-dimensional normed space then the weak topology on Xis strictly coarser than the norm
topology.
Proof. Each f∈X∗is continuous for the norm topology, and so this topology if finer than the weak topology (by the definition of
σ(X, X ∗) as the coarsest topology making all f∈X∗continuous.
The unit sphere SX={x∈X:kxk= 1}is closed in the norm topology. I shall show that SXis not closed in the weak topology
whenever Xis infinite-dimensional. By Lemma 5.2, every weak neighbourhood Uof 0 contains a subspace X1of Xwhich is of
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